Geometric Distribution

The geometric distribution tells us how likely the first success is given a sequence of \(n\) trials of independent Bernoulli events with probability \(p\). he formula for the Probability Mass Function (PMF) of the random variable \(\mathbf{X}\) is :

\[ \mathbf{X} \sim \text{Geometric}(p) \]

Since this calculates the probability of the first success occurring on the \(n^th\) trials, the previous trials (before the \(n^th\)) must be failures. By the independence of the trials, the probability of this event is:

\[ \begin{align*} P(\mathbf{X} = n) &= P(X_0 = 0 \cap X_1 = 0 \cap \cdots \cap X_{n-1} = 0 \cap X_n = 1)\\ &= P(X_0 = 0) \cdot P(X_1 = 0) \cdots P(X_{n-1} = 0) \cdot P(X_n = 1)\\ P(\mathbf{X} = n)&=(1-p)^{n-1}p \end{align*} \]